Finite Group Databasebeta

← #10944163

Subgroups — #10944163

Order 768.

1590 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C2×#5241C212yesmaximal, complemented
3#82526C212yesmaximal, complemented
4#82378C211yesmaximal, characteristic, complemented
5#71667C214yesmaximal, complemented
6#71650C212yesmaximal, complemented
7#71646C212yesmaximal, complemented
8#71617C211yesmaximal, characteristic, complemented
9#32920—31nomaximal, Sylow
10C2×C3×H(5,𝔽2)C2211yescharacteristic
11C3×#520—22no—
12C3×#520C2212yes—
13C3×#518—22no—
14C3×#518—22no—
15C3×#518C2211yescharacteristic
16C3×C4×#41—22no—
17C3×C4×#41C2212yes—
18C2×C3×C4×D4—22no—
19C2×C3×C4×D4—22no—
20C2×C3×C4×D4C2211yescharacteristic
21C2×#949C2214yes—
22C2×#940C2212yes—
23C2×#930—22no—
24C2×#874C2211yescharacteristic, complemented
25#5241—28no—
26C2×#837C2214yes—
27C2×#833C2212yes—
28#5027—22no—
29#5027—22no—
30#5015—22no—
31#5015—22no—
32#5001C2214yescomplemented
33#4988C2212yescomplemented
34#4960—22no—
35C4×C4⋊Dic3C2211yescharacteristic, complemented
36#4924C2212yescomplemented
37#4532C2214yescomplemented
38#4530C2212yescomplemented
39C4×H(5,𝔽2)D311yescharacteristic, complemented
40#3038—32no—
41C2×#453—32no—
42#2541—31no—
43#1887—32no—
44#1883—34no—
45#1806—32no—
46#1800—31no—
47C3×H(5,𝔽2)—24no—
48C3×H(5,𝔽2)—22no—
49C3×H(5,𝔽2)D414yescomplemented
50C2×C3×#41—22no—
51C2×C3×#41C2312yes—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—22no—
56C23×C3×C4D412yes—
57C22×#67—24no—
58C22×D12—21no—
59C22×D12—21no—
60C22×D12C2312yes—
61C3×C4×Q8—22no—
62C3×C4×Q8D412yescomplemented
63C3×C4×Q8D412yescomplemented
64C3×C4×D4—24no—
65C3×C4×D4—24no—
66C3×C4×D4—24no—
67C3×C4×D4—24no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C3×C4×D4—24no—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C3×C4×D4D412yescomplemented
77C3×C4×D4D412yescomplemented
78C3×#117—24no—
79C3×#117—22no—
80C3×#117—22no—
81C3×#117C2312yes—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—24no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4D412yes—
89C2×C3×C42—22no—
90C2×C3×C42—22no—
91C2×C3×C42C2311yescharacteristic
92#949—28no—
93#949—28no—
94C2×#215—24no—
95#940—24no—
96#940—24no—
97C2×#213—24no—
98#930—48no—
99C2×(C2×D3)⋊C4—24no—
100C2×C4⋊Dic3—21no—
101C2×C4×Dic3C2312yes—
102C2×#208C2314yes—
103#874—22no—
104#874—22no—
105#837—28no—
106#833—24no—
107C3⋊#95—21no—
108#809—24no—
109#809—24no—
110C4⋊(C3⋊C8)—22no—
111C4⋊(C3⋊C8)—22no—
112C2×H(5,𝔽2)C2×D311yescharacteristic
113C2×AGL(1,ℤ/8)—34no—
114C2×#127—32no—
115C2×#127—31no—
116C2×#121—62no—
117#520—22no—
118#520C2×D312yes—
119#518—22no—
120#518—22no—
121#518C2×D311yescharacteristic
122C4×#41—22no—
123C4×#41C2×D312yes—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4C2×D311yescharacteristic
127#460—62no—
128#460—62no—
129#459—62no—
130#459—62no—
131#453—68no—
132#423—32no—
133C2×#104—34no—
134#418—34no—
135C2×#100—32no—
136#392—62no—
137#390—32no—
138C4×C4⋊C4—31no—
139#343—32no—
140#339—34no—
141C24×C3—22no—
142C24×C3C2×D412yes—
143C23×D3—22no—
144C3×#41—28no—
145C3×#41—28no—
146C3×#41—22no—
147C3×#41—22no—
148C3×#41C2×D414yes—
149C3×#41#4114yes—
150C2×C3×Q8C2×D412yes—
151C2×C3×D4—24no—
152C2×C3×D4—28no—
153C2×C3×D4—28no—
154C2×C3×D4—28no—
155C2×C3×D4—22no—
156C2×C3×D4—28no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4C2×D412yes—
160C2×C3×D4C2×D412yes—
161C2×C3×D4#4112yes—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—44no—
172C22×C3×C4—84no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C2×#67—48no—
178C2×#67—48no—
179C2×#67—84no—
180C22×Dic3—22no—
181C2×D12—24no—
182C2×D12—22no—
183C2×D12—22no—
184C2×D12—24no—
185C2×D12—44no—
186C2×D12—41no—
187C2×D12—42no—
188C2×D12—41no—
189C2×D12—82no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4C2×D412yes—
195C3×C4⋊C4C2×D412yes—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—22no—
201C3×C22⋊C4—24no—
202C3×C42—24no—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42C2×D412yes—
207C3×C42C2×D412yes—
208#215—48no—
209#213—48no—
210(C2×D3)⋊C4—48no—
211C4⋊Dic3—24no—
212C4×Dic3—22no—
213C4×Dic3—22no—
214#208—24no—
215#208—24no—
216C2×C3⋊C8—24no—
217H(5,𝔽2)—24no—
218H(5,𝔽2)—22no—
219H(5,𝔽2)D1214yescomplemented
220C2×#41—22no—
221C2×#41C22×D312yes—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—32no—
225C22×D4—64no—
226C22×D4—61no—
227C22×D4—61no—
228C22×D4C22×D311yescharacteristic
229C23×C4—22no—
230C23×C4#6712yes—
231AGL(1,ℤ/8)—68no—
232AGL(1,ℤ/8)—68no—
233C2×SD16—64no—
234C2×D8—64no—
235C2×M16—34no—
236#127—64no—
237#127—64no—
238#127—62no—
239#127—62no—
240#121—128no—
241C4×Q8—22no—
242C4×Q8#6712yescomplemented
243C4×Q8#6712yescomplemented
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—24no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4#6712yescomplemented
257C4×D4#6712yescomplemented
258#117—24no—
259#117—22no—
260#117—22no—
261#117C22×D312yes—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4C22×D311yescharacteristic
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—64no—
270C2×C22⋊C4#6712yes—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—32no—
274C2×C42C22×D311yescharacteristic
275C4⋊C8—62no—
276C4⋊C8—62no—
277#104—68no—
278#102—64no—
279#102—64no—
280#100—64no—
281#95—61no—
282C23×C3—28no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—82no—
289C23×C3—82no—
290C23×C3#12114yes—
291C23×C3C22×D411yescharacteristic
292C22×D3—44no—
293C22×D3—44no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C3×Q8—22no—
298C3×Q8#12114yescomplemented
299C3×D4—216no—
300C3×D4—24no—
301C3×D4—24no—
302C3×D4—24no—
303C3×D4—22no—
304C3×D4—24no—
305C3×D4#12114yescomplemented
306C2×C3×C4—28no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—41no—
315C2×C3×C4—44no—
316C2×C3×C4—44no—
317C2×C3×C4—42no—
318C2×C3×C4—82no—
319C2×C3×C4—82no—
320C2×C3×C4—82no—
321C2×C3×C4—84no—
322C2×C3×C4—82no—
323C2×C3×C4#12012yes—
324C2×C3×C4#12011yescharacteristic
325C2×C3×C4#12011yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D412yes—
328C2×C3×C4C2×#4112yes—
329C2×C3×C4C2×#4111yescharacteristic
330#67—816no—
331C2×Dic3—24no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334D12—44no—
335D12—44no—
336D12—82no—
337D12—84no—
338D12—82no—
339C3⋊C8—44no—
340C24—22no—
341C24—62no—
342C24C2×#6712yes—
343#41—28no—
344#41—28no—
345#41—22no—
346#41—22no—
347#41C2×D1214yes—
348#41#23514yes—
349C2×Q8C2×#6712yes—
350C2×D4—24no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—22no—
355C2×D4—28no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—64no—
361C2×D4—64no—
362C2×D4—124no—
363C2×D4—128no—
364C2×D4—128no—
365C2×D4—122no—
366C2×D4—121no—
367C2×D4—121no—
368C2×D4—242no—
369C2×D4—244no—
370C2×D4C2×D1212yes—
371C2×D4#23512yes—
372C2×D4C2×#6712yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—44no—
383C22×C4—62no—
384C22×C4—84no—
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6712yes—
387C22×C4C2×#6711yescharacteristic
388C22×C4C23×D311yescharacteristic
389SD16—128no—
390D8—128no—
391M16—64no—
392M16—64no—
393C2×C8—64no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—64no—
399C4⋊C4C2×#6712yes—
400C4⋊C4C2×#6712yes—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—24no—
407C22⋊C4—128no—
408C42—24no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—62no—
413C42—62no—
414C42C2×#6712yes—
415C42C2×#6712yes—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—44no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—82no—
426C22×C3—88no—
427C22×C3C2×#12011yescharacteristic
428C22×C3C2×#12111yescharacteristic
429C22×C3#52511yescharacteristic
430C2×D3—84no—
431C2×D3—84no—
432C2×D3—82no—
433C2×D3—82no—
434C2×D3—82no—
435C3×C4—24no—
436C3×C4—21no—
437C3×C4—22no—
438C3×C4—42no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4#45312yescomplemented
442C3×C4C2×#12112yes—
443C3×C4#52512yes—
444Dic3—44no—
445C23—28no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—42no—
450C23—42no—
451C23—82no—
452C23—82no—
453C23—124no—
454C23—124no—
455C23—122no—
456C23—122no—
457C23—122no—
458C23#93014yes—
459C23C22×#6711yescharacteristic
460Q8—22no—
461Q8#93014yescomplemented
462D4—216no—
463D4—24no—
464D4—24no—
465D4—24no—
466D4—24no—
467D4—22no—
468D4—124no—
469D4—124no—
470D4—242no—
471D4—244no—
472D4—2416no—
473D4—242no—
474D4#93014yescomplemented
475C2×C4—28no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—64no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—84no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4#95312yes—
496C2×C4#95311yescharacteristic
497C2×C4#95311yescharacteristic
498C2×C4C22×D1211yescharacteristic
499C2×C4C2×#23512yes—
500C2×C4C2×#23511yescharacteristic
501C2×C4C22×#6712yes—
502C8—124no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—44no—
506C2×C3—44no—
507C2×C3#284711yescharacteristic
508C2×C3C2×#45311yescharacteristic
509C2×C3#302111yescharacteristic
510D3—162no—
511C22—21no—
512C22—21no—
513C22—24no—
514C22—24no—
515C22—44no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—82no—
521C22—88no—
522C22—244no—
523C22—244no—
524C22—242no—
525C22—242no—
526C22—242no—
527C22C2×#93011yescharacteristic
528C22#579111yescharacteristic
529C22C2×#95311yescharacteristic
530C4—24no—
531C4—21no—
532C4—22no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—124no—
537C4#524112yescomplemented
538C4C2×#93012yes—
539C4#579112yes—
540C3#3292011yescharacteristic, complemented, Sylow
541C2—21no—
542C2—21no—
543C2—44no—
544C2—44no—
545C2—482no—
546C2#8403211yescharacteristic
547C2C2×#524111yescharacteristic
548C2#8412211yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×H(5,𝔽2)C2
socleC22×C3C2×#120
hypercenterC4×H(5,𝔽2)D3
hypocenterC3#32920
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×H(5,𝔽2)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×C4×H(5,𝔽2)C2semidirect
2. C2×#5241C2semidirect
3. #82526C2semidirect
4. #82378C2semidirect
5. #71667C2semidirect
6. #71650C2semidirect
7. #71646C2semidirect
8. #71617C2semidirect
24. C2×#874C22semidirect
32. #5001C22semidirect
33. #4988C22semidirect
35. C4×C4⋊Dic3C22semidirect
36. #4924C22semidirect
37. #4532C22semidirect
38. #4530C22semidirect
39. C4×H(5,𝔽2)D3semidirect
49. C3×H(5,𝔽2)D4semidirect
62. C3×C4×Q8D4semidirect
63. C3×C4×Q8D4semidirect
76. C3×C4×D4D4semidirect
77. C3×C4×D4D4semidirect
219. H(5,𝔽2)D12semidirect
242. C4×Q8#67semidirect
243. C4×Q8#67semidirect
256. C4×D4#67semidirect
257. C4×D4#67semidirect
298. C3×Q8#121semidirect
305. C3×D4#121semidirect
441. C3×C4#453semidirect
461. Q8#930semidirect
474. D4#930semidirect
537. C4#5241semidirect
540. C3#32920semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×C4×H(5,𝔽2)2
C2×#52412
#825262
#823782
#716672
#716502
#716462
#716172
#329203